Backpropagating Pauli Propagation

In the context of quantum circuit optimization, how can one compute parameter gradients efficiently without the prohibitive memory overhead associated with storing all intermediate states in reverse-mode automatic differentiation? The authors seek a method that leverages the sparse Pauli dynamics (SPD) framework to enable scalable optimization for large-scale quantum simulations.

The authors propose a backpropagation algorithm that exploits the inherent reversibility of quantum circuits. Instead of caching the full forward trajectory of operators, the algorithm performs a forward sweep to reach the final state, initializes the adjoints of the cost function, and then executes a backward sweep. During this backward pass, the algorithm recomputes the necessary intermediate operators on the fly using the inverse of the Pauli update rules. This approach effectively mirrors the structure of reverse-mode automatic differentiation but avoids the memory bottleneck by trading it for a recomputation step.

The proposed method achieves a memory reduction of O(n_param) compared to conventional reverse-mode automatic differentiation, where n_param is the number of parameterized gates. The computational complexity remains comparable to standard sparse Pauli simulation techniques. Numerical demonstrations on transverse-field Ising models (up to 3D) and 3D Heisenberg models show that the gradient accuracy remains consistent with the accuracy of the underlying expectation value estimates. The authors demonstrate that this technique is highly effective for tasks such as low-energy state preparation and time-evolution circuit compression.

Classical optimization of quantum circuits is essential for variational algorithms, yet memory limitations often restrict the size of circuits that can be optimized. By enabling gradient-based optimization with significantly lower memory requirements, this method allows researchers to tackle larger and more complex quantum systems, facilitating more accurate state preparation and more efficient circuit design in quantum simulation applications.